nLab
coadjoint orbit

Context

Group Theory

Representation theory

Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Definition

Given a Lie group GG, it acts smoothly on the dual g *g^* of its Lie algebra gg by the coadjoint action. The orbits of that action are called coadjoint orbits.

Coadjoint orbits are especially important in the orbit method of representation theory or, more generally, geometric quantization.

Sometimes coadjoint orbits are studied in the infinite-dimensional case (for example in study of Virasoro algebra).

Properties

As symplectic leafs of the Lie-Poisson structure

The dual g *g^* of a (say finite-dimensional real) Lie algebra has a structure of a Poisson manifold with the Poisson structure due to A. Kirillov and Souriau, called the Lie-Poisson structure, namely for any ag *a\in g^*,

{f,g}(a):=[df a,dg a],a \{ f, g\}(a) := \langle [d f_a, d g_a],a\rangle

The coadjoint orbits are the symplectic leaves of that structure; hence each orbit is a symplectic manifold.

References

  • B. C., The Structure of the Space of Coadjoint Orbits of an Exponential Solvable Lie Group, ransactions of the American Mathematical Society Vol. 332, No. 1 (Jul., 1992), pp. 241-269, (JSTOR)

Revised on October 29, 2013 23:46:24 by Urs Schreiber (82.169.114.243)